Logan Nielsen
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Note · Quantitative Finance

Benchmark Portfolio Optimization

Optimizing a portfolio relative to a respected benchmark.

June 15, 2021 · 4 min read

Introduction

We believe we can do better than a benchmark portfolio our client would be 100% invested in if our fund didn’t exist. We’re going to make a portfolio that beats the benchmark on a alpha basis.

There are two ways we can approach the problem.

  1. Maximizing the information ratio.
  2. Benchmark hedging.

The Portfolios

We want to hold some active portfolio, AA, with asset weights h\pmb h which tracks some benchmark, BB, with weights b\pmb b. We denote the weights of the active portfolio by h\pmb h and the benchmark as b\pmb b. We seek to find the optimal portfolio, WW, with weights, w=h+b\pmb w= \pmb h + \pmb b. Our final portfolio will be fully invested, w′1\pmb w'\pmb 1=1, and the benchmarks, by definition, much sum to one as well, b′1=1\pmb b' \pmb 1 =1. This implies that the weight of our active holding sum to zero, h′1=0\pmb h'\pmb 1=0.

Maximizing the Information Ratio

The information ratio is defined as:

IR=E[rA]−E[rP]TE\begin{align} IR = \frac{E[r_A]-E[r_P]}{TE} \end{align}

Where TETE is tracking error and defined as:

TE=Var(rA−rB)=Var(w−b)′r=Var(h′r)=h′Vh\begin{align} TE &= \sqrt{Var(\pmb r_A-\pmb r_B)} \\ &= \sqrt{Var(\pmb w-\pmb b)'\pmb r} \\ &= \sqrt{Var(\pmb h'\pmb r)} \\ &= \sqrt{\pmb h'V\pmb h} \end{align}

Constraints

  1. We want to set a cap how much tracking error, σˉ\bar \sigma, we’ll allow. Therefore h′Vh≤Var(σˉ)\sqrt{\pmb h'V \pmb h} \leq Var(\bar \sigma).
  2. We also want βA,B=1\beta_{A,B} = 1 meaning that we’re not taking an additional risk relative to the benchmark. This constraint is equivalent to βU,B′h=0\pmb \beta_{\pmb U,B}'\pmb h= 0. Note that βA,B=Cov(rA,rB)Var(rB)\beta_{A,B} = \frac{Cov(r_A, r_B)}{Var(r_B)} while βU,B\pmb \beta_{\pmb U,B} is a vector of the betas for all assets that can be selected from with the benchmark.

 \qquad\ If you need help seeing the equivalence mentioned:

1=w′βU,B=(h+b)′βU,B=h′βU,B+b′βU,B=h′βU,B+1  ⟹  h′βU,B=0\begin{align} 1 &= \pmb w'\pmb \beta_{\pmb U,B} \\ &= (\pmb h + \pmb b)'\pmb \beta_{\pmb U,B} \\ &= \pmb h'\pmb \beta_{\pmb U,B} + \pmb b'\pmb \beta_{\pmb U,B} \\ &= \pmb h'\pmb \beta_{\pmb U,B} + 1 \\ \implies \pmb h'\pmb \beta_{\pmb U,B} &= 0 \end{align}

Setup

Turns out that h′α=E[rA]−E[rB]\pmb h' \pmb \alpha = E[r_A] - E[r_B] given our constraints.

Therefore we can write the lagrangian as:

max⁡hh′α−λh′Vh−γ′(Ch−c)\begin{align} \max_{\pmb h} \pmb h'\pmb \alpha - \lambda \pmb h' V \pmb h - \pmb \gamma'(C\pmb h - \pmb c) \end{align}

Where our constraints are:

C=[1′β′],  c=[00]\begin{align} C= \begin{bmatrix} \pmb 1' \\ \pmb \beta' \end{bmatrix}, \; \pmb c= \begin{bmatrix}0 \\ 0\end{bmatrix} \end{align}

After solving (insert link to solving the optimal portfolio we find that

h~=(2λ)−1V−1[I−C′(CV−1C′)−1CV−1]α+V−1C′(CV−1C′)−1c\begin{align} \pmb{\tilde h} = (2\lambda)^{-1}V^{-1}\big[I - C'(CV^{-1}C')^{-1}CV^{-1}\big]\pmb \alpha + V^{-1}C'(CV^{-1}C')^{-1}\pmb c \end{align}

Finally, we scale our answer to the appropriate risk, σˉ\bar \sigma. We want to find kk such that:

(kh~)′V(kh~)=∣k∣h~′Vh~=σˉ  ⟹  k=σˉh~′Vh~=σˉσH~\begin{align} \sqrt{(k\pmb{\tilde h})'V(k\pmb{\tilde h})}= |k|\sqrt{\pmb{\tilde h}'V\pmb{\tilde h}}&= \bar \sigma \\ \implies k &= \frac{\bar \sigma}{\sqrt{\pmb{\tilde h}'V\pmb{\tilde h}}} \\[12pt] &=\frac{\bar \sigma}{\sigma_{\tilde{H}}} \end{align}

Finally, h∗=k⋅h~\pmb h^* = k \cdot \pmb {\tilde h}

Benchmark Hedging

Our second strategy for beating the benchmark is called benchmark hedging. This is employed when we have predictions about individual asset alphas in the portfolio.

We will create a portfolio that is a mix of the stocks for which we have VV views given by α∈RV×1\pmb \alpha \in \mathbb R^{V\times 1} and the ETF benchmark b∈RN×1\pmb b \in \mathbb R^{N \times 1}.

We construct α=[αV0]′∈R(V+1)×1\pmb \alpha = \begin{bmatrix}\pmb \alpha_V \\ 0\end{bmatrix}' \in \mathbb R^{(V+1)\times 1}, V=[VV(Vb)V(Vb)V′bVb]∈R(V+1)×(V+1)V = \begin{bmatrix}V_V & (V\pmb b)_V \\ (V\pmb b)_V' & \pmb b V\pmb b\end{bmatrix} \in \mathbb R^{(V+1)\times(V+1)}, and h=[hVhETF]∈R(V+1)×1\pmb h = \begin{bmatrix} \pmb h_V \\h_{ETF} \end{bmatrix} \in \mathbb R^{(V+1) \times 1}.

For the constraints:

C=[1V′1βV′1],  c=[00]\begin{align} C = \begin{bmatrix} \pmb 1'_V & 1 \\ \pmb \beta_V' & 1 \end{bmatrix}, \; \pmb c = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \end{align}

Now we can solve:

max⁡hh′α−λh′Vh−γ′(Ch−c)\begin{align} \max_{\pmb h} \pmb h'\pmb \alpha - \lambda \pmb h' V \pmb h - \pmb \gamma'(C\pmb h - \pmb c) \end{align}

Where we rescale the holdings to meet the target risk σˉ\bar \sigma.